Signaling Without Handicap

Transcription

1 Signaling Without Handicap Joel Sobel December 5, 2007 Department of Economics, University of California, San Diego, I base this paper on a presentation made at the conference Sex and the Signal. I am grateful to the organizers for inviting me to the conference and to the participants (in particular, Gadi Katzir, Tamar Keasar, Michael Lachmann, Avi Shmida, and Frank Thuijsman) for encouragement. I thank the Guggenheim Foundation, NSF, and the Secretaría de Estado de Universidades e Investigación del Ministerio de Educación y Ciencia (Spain) for financial support. I am grateful to the Departament d Economia i d Història Econòmica and Institut d Anàlisi Econòmica of the Universitat Autònoma de Barcelona for hospitality and administrative support.

2 I Introduction The handicap principal explains how observable differences in features of plants or animals may serve as signals. Informative signaling is possible if the marginal cost of sending a signal is decreasing with fitness and there are rewards associated with being seen as more fit (typically in the form of greater opportunities to reproduce). Handicaps are costly. Animals engage is costly signaling because the benefits outweigh these costs. I want to point out circumstances under which communication is possible without cost. I make this observation using a simple model of signaling. Section II describes this model. In a signaling game, one player has private information. Section III describes two kinds of necessary conditions for effective cheap talk. These conditions imply there must be some common interest a desire to coordinate under certain circumstances in order for communication to be valuable. Specifically, nontrivial communication via cheap talk requires that there be some conflict of interest between different signaler. If the signaler always wanted his message to lead to the same action, then he would always send the same message. 1 Nontrivial communication also requires some common interest between the animal who signals and the animal who responds, because otherwise no message would be credible. Section IV describes examples in which the necessary conditions from Section III hold and demonstrates the possibility of effective communication without handicap. In one example, a common interest between informed and uninformed player arises because participants are costly related. In the other, the common interest is due to advantages from coordination. In both cases, changes in the characteristic or private information of the informed player exist because the information directly influences preferences. Section V points out that the necessary conditions of Section III are weak and suggests that a detailed description of many strategic environments will reveal the possibility of coordination. When either the signal or the action is multidimensional, more opportunities for effective communication arise. Section VI argues that even in situations where signaling must be costly to some in order to be effective, it may be that those who choose signal can signal cheaply. Section VII is a brief conclusion. 1 This statement must be modified because if the signaler is indifferent between two actions, he might send an informative message. 1

3 II Basic Model This section describes the basic signaling game. There are two players, called S (for Sender) and R (for Receiver). S knows the value of some random variable t whose support is a given set T. t is called the type of S. The prior beliefs of R are given by a probability distribution π( ) over T ; these beliefs are common knowledge. When T is finite, π(t) is the prior probability that the sender s type is t. When T is uncountably infinite, π( ) is a density function. Player S, learns t, sends to R a signal m, drawn from some set M. Player R receives this signal, and then takes an action a drawn from a set A. This ends the game: The payoff to i is given by a function u i : T M A R. Defining Nash equilibrium for the basic signaling game is completely straightforward when T, S, and A are finite sets. In this case a behavior strategy for S is a function µ : T M [0, 1] such that m M µ(t, m) = 1 for all t. µ(t, m) is the probability that sender-type t sends the signal m. A behavior strategy for R is a function α : M A [0, 1] where a A α(m, a) = 1 for all m. α(m, a) is the probability that R takes action a following the signal m. Behavior strategies (α, µ ) form a Nash Equilibrium if and only if for all t T µ(t, m) > 0 implies a A U S (t, m, a)α(m, a) = max m M U S (t, m, a)α(m, a) (1) and, for each m M such that µ(t, m)π(t) > 0 and if µ(t, m)π(t) > 0, (2) t T t T then α(m, a) > 0 implies t T where U R (t, m, a)β(t, a) = max a A β(t, m) = a A U R (t, m, a )β(t, a ) t T (3) µ(t, m)π(t) t T µ(t, m)π(t ). (4) Condition (1) states that S places positive probability only on signals that maximize expected utility. This condition guarantees that S responds optimally to R s strategy. Condition (3) states that R places positive probability 2

4 only on actions that maximize expected utility, where is taken with respect to the distribution β(, m) following the signal m. Condition (4) states that β(, m) accurately reflects the pattern of play. It requires that R s beliefs be determined using S s strategy and the prior distribution whenever possible. Nash equilibrium is a necessary condition for stability in signaling games. It may be that only a subset of equilibrium strategies are evolutionarily stable. In this paper, I focus on pure strategies. The results continue to hold for mixed-strategy equilibria, although the evolutionary stability of mixed strategies is in doubt. This canonical game captures the essential features of the classic applications of signaling in both biology and economics. In the classical applications, signaling is costly, the cost function satisfies a property that makes the marginal cost of signaling decreasing in t and the best response to higher t is an action that is more attractive for all types of Sender. For example, it may be that u S (t, m, a) = V (a) C(t, m) and u R (t, m, a) = (a t) 2, where V ( ) is strictly increasing and C(t, m) C(t, m ) is increasing in t when m < m. Under these conditions, when a Sender is indifferent between two messages, a higher type of Sender will strictly prefer to send the higher message. Further, each Sender type wishes to be perceived as the highest type possible. In Zahavi [8] s formulation, C( ) is the handicap function. We say that the signaling game is a cheap-talk game if u i (t, m, a) does not depend on m. That is, the message selected has no direct influence on payoffs. That is, u i (t, m, a) = U i (t, a). Of course, if different messages lead to different actions, then the message will have an indirect influence on payoffs. In cheap-talk games, no type has a comparative advantage in sending signals. There are no handicaps, so signaling cannot operate as it does in standard models. There is still scope for non-trivial communication in interesting situations. III Necessary Conditions for Effective Talk An equilibrium exhibits effective communication if the Receiver takes more than one action with positive probability. In an equilibrium with effective communication, the Receiver draws some inference from communication and therefore (because the Receiver takes a best response to signals) is better off than if communication was not feasible. Effective communication may arise when the Sender does not care about the Receiver s action and so is 3

5 willing to provide information freely. Effective communication is non trivial if some Sender type strictly prefers to induce one equilibrium action to another. Communication is non-trivial if the Sender s message actually can change his payoff. It is useful to think about cheap-talk equilibria in terms of the set of actions induced, that is the set of actions that are taken with positive probability in equilibrium. There is no communication when the set of induced actions contains only one element. If communication is free, then there always exists a babbling equilibrium in which the message of the Sender conveys no information and the Receiver takes the same action no matter what the Sender says. This assertion follows immediately from the definitions: take the Receiver s strategy to be constant and equal to the action that maximizes U R (t, a)π(t) (5) t T and let the Sender send a particular message, m, with probability one. It is apparent that if the Receiver s action does not depend on m, then the Sender best responds by sending m. On the other hand, if the Sender always sends m, then the Receiver does best by taking an action defined by (5). In this equilibrium, the Receiver stubbornly refuses to listen to the Sender and the Sender, confronted with an unresponsive audience, can do no better than supply no information (and hence justify the Receiver s stubbornness). This babbling outcome may be dynamically unstable, but moving from the equilibrium requires the two players to gain an understanding that new messages can have meanings different from the equilibrium message. There is no guarantee that this is possible. I will show that it is possible to communicate effectively without handicaps. Before I do so, I point out two conditions that are necessary for non-trivial cheap-talk communication. III.1 Conflict Between Sender Types In standard applications of signaling games, preferences over actions are independent of type. That is, u S (t, m, a) > u S (t, m, a ) if and only if u S (t, m, a ) > u S (t, m, a ). Independence follows if the action is a monetary payoff (for economics) or the number of offspring (biology). In these settings, there is no possibility for effective cheap-talk communication. 4

6 In a cheap-talk game the sender s preferences induce a preference ranking over actions. The type t Sender strictly prefers action a to a if and only if U S (t, a) > U S (t, a ) and is indifferent if and only if U S (t, a) = U S (t, a ). Proposition 1. There can be no non-trivial communication in a cheap-talk game in which Sender s preferences over actions are independent of t. Proof. Suppose there is an equilibrium which non-trivial communication. This means that there are messages m and m that are both sent with positive probability in equilibrium and actions, such that α(m) α(m ) and a t such that U S (t, α(m)) > U S (t, α(m )). (6) If the Sender s preferences are independent of t, then inequality (6) implies that no Sender type will send m with positive probability, which is a contradiction. I emphasize that it is possible to have effective communication if the Sender s preferences over actions are independent of t. For example, if the Sender does not care what action the Receiver takes, then there is an equilibrium in which the Sender signals honestly. Proposition 1 implies unless there is a conflict of interest between Sender types there cannot be cheap-talk communication that benefits the Sender. Of course, if the Sender is never indifferent between possible actions, then any effective communication will be non-trivial. The necessary condition for communication appears strong and naively rules out cheap talk as a way for hungry birds to demand food from their mother (because all birds prefer to have more food), flowers that advertise for a good pollinator (because all flowers want to be pollinated), or signals that captivate potential mates. In Section V, I will argue that the condition is less restrictive than in appears to be. III.2 Common Interest Between Sender and Receiver If R and S have common preferences, then one player favors an action a over an action a, then so does that other. That is, U S (t, a) > U S (t, a ) if and only if U R (t, a) > U R (t, a ). (7) 5

7 If their interests are completely opposed, then U S (t, a) > U S (t, a ) if and only if U R (t, a) < U R (t, a ). (8) Condition (7) holds in coordination games. If the condition holds for all t, it is apparent that there is a fully revealing cheap-talk equilibrium. If different types send different messages, then R infers t from the message and is able to take the action that is best for both players. Condition (8) holds in zero-sum games. Complete conflict rules out the possibility of meaningful communication. Informative cheap talk is possible even if (7) does not hold, as subsequent examples demonstrate. In this section I show that weaker versions of (8) limit the extent of communication. First, I make precise the way in which (8) conflict limits communication. Proposition 2. If U R (t, a) > U R (t, a ) if and only if U S (t, a) < U S (t, a ) and U R (t, a) < U R (t, a ) if and only if U S (t, a) > U S (t, a ), then in any equilibrium, S and R receive payoffs attainable in a babbling equilibrium. Proof. Fix an equilibrium. Suppose that there is a type t and a pair of actions induced in equilibrium such that U S (t, a) > U S (t, a ). Let m be such that α(m) = a and let T (m) be the set of types that use m. It follows that U S (t, a) > U S (t, a ) for all t T (m) with strict inequality for at least one t. By assumption, U R (t, a ) > U R (t, a) for all t T (m) with strictly inequality for at least one t. Hence a is a better response to m for R than a, which is a contradiction. It follows that every action induced in equilibrium gives S the same utility. By assumption, R must be indifferent over all actions induced independent of S s type, so any action taken with positive probability in equilibrium is a pooling action. There are other conditions ways in which preferences of S and R can differ so much that effective communication is not possible. Let a R (t) denote R s best response to t. That is, a R (t) is the solution to: max a A U R (a, t). Assume for convenience that the maximizer is unique. I will say that R can learn t if there exists an equilibrium with effective communication in which t sends a message m with the property that a(m) = a R (t). If R cannot learn t, then every equilibrium is equivalent to an equilibrium in which t pools with 6

8 other types. On the other hand, just because R can learn t (according to the definition), does not mean that R will learn t. For example, if a R (t) is the ex ante optimal action for R, then there is a babbling equilibrium in which all messages induce the action a R (t). In this case, R will take the action a R (t) (and t will induce this action), but R will learn nothing from S s message. Proposition 3. R can learn t only if there exists a, such that U S (t, m, a R (t)) U S (t, m, a). Sender type t can only be made to reveal his type truthfully if there is no way to induce an action better that a R (t). Proof. Suppose there is a t such that U S (t, a) > U S (t, a R (t)) for all a a R (t). The only way in which t will send a message m such that a(m) = a R (t) is if all messages lead to this action. A corollary of Proposition 3 is that there cannot be an equilibrium with effective communication if for all t, U S (t, a) > U S (t, a R (t)) for all a a R (t). If one imposes more structure on preferences, a stronger result is possible. Suppose that U i is twice continuously differentiable and strictly concave in action a. Assume further that U S has a negative cross partial, while U R has a positive cross partial. These conditions guarantee that a R is strictly increasing while a S is strictly decreasing. These conditions hold in the special case U R (t, a) = (a t) 2 and in the model of Crawford and Sobel [3]. Proposition 4. If a R is strictly increasing and a S is strictly decreasing, then there can be no effective communication in equilibrium. Proof. Suppose that there is an equilibrium in which the messages m and m are sent with positive probability, and that these messages lead to the actions a > a. By continuity, there is a t that is indifferent between the two actions. No Sender type higher than t induces the action a, so a a R (t ). (9) 7

9 No Sender type lower than t induces the higher a. Hence a a R (t ). (10) Inequalities (9) and (10) imply that a a, which is a contradiction. Under the conditions of Proposition 4 it is possible that there is a t such that a S (t) = a R (t), so that one type of Sender has identical interests to the Receiver. In general, the condition in Proposition 3 could fail for most types. Nevertheless, communication breaks down. What makes communication impossible in this case is that whenever R prefers a to a the Sender has the opposite preference. Proposition 4 provides a necessary condition for effective communication. The condition is not sufficient. Crawford and Sobel give a condition that rules out effective communication even when a i is increasing for i = S or R. A naive interpretation of the results in this section imply that cheap talk will have no impact in strategic interactions between predator and prey, because the interests of the two parties are completely opposed. More generally, there can be no effective cheap talk when the game between S and R is constant sum. IV Examples of Effective Communication I argued that cheap talk will generally not be useful unless there is some conflict of interest between Sender types and some common interest between the Sender and the Receiver. These observations are merely an abstract way to say that in order for costless signaling to be effective, there must be some circumstances in which the Sender and Receiver want to coordinate. Coordination problems arise naturally. In this section I describe two examples of communication via cheap talk. IV.1 Communication Between Family Members Maynard Smith [6] introduced this game. Bergstrom and Lachmann [1] pointed out that it may be a better model of cheap communication. Here is the story (simplified somewhat). There is a parent and a child. The health of the parent is y and known to both parent and child. The health 8

10 of the child is t and is known perhaps to the child but not to the parents. Since the parent and child are related, they have mutual interests. They are related so they have some common interest: the health of the child enhances the reproductive success of the parent and, similarly, the health of the parent enhances the degree to which the child s genes appear in future generations. On the other hand, the parent and child have some conflict. In the example, assume that the choice of the parent is whether to feed the child or keeps food for herself. The health of the player who gets to eat increases to one. The health of the other player remains at its original level. In terms of the formal model, the informed child in the Sender. His type t, which is private information to the Sender and drawn from a density f( ) supported on [0, 1]. After observing his type, the Sender sends a message m to the parent, who acts as Receiver. The parent must then decide whether to transfer a resource to the child. If the parent transfers the resource, the child s direct benefit is 1 while the parent s direct benefit is y (0, 1). If the parent does not transfer the resource, the child s direct benefit is t while the parent s direct benefit is 1. Total fitness is the weighted sum of a player s direct benefit and the benefit of the other, weighted by k (0, 1]. k is the degree to which the players are related. 2 Consequently, if a transfer is made with probability 1 a, then U S (t, a) = (1 a)(1 + ky) + a(t + k) while U R (t, a) = a(1 + kt) + (1 a)(y + k). All aspects of the model except t are common knowledge. This model does not satisfy the strict concavity assumption of the basic model of Crawford and Sobel [3], but otherwise is analogous, and it shares the property that optimal complete-information actions are (weakly) increasing in t. Provided that y + k > 1, both players benefit from (resp. are hurt by) transfers when t is low (resp. high), but the child prefers transfers for more values of t than the parent. The child likes weakly lower values of a than the parent for all t. This game has a babbling equilibrium in which there is no effective communication. In the babbling equilibrium, the parent makes a transfer if 1 + k t < y + k, where t is the expected value of t. Otherwise she makes no transfer. That is, the parent ignores whatever information may be in the message of the child 2 In an economic context, k could be viewed as an altruism parameter. 9

11 and makes a transfer if t is expected to be low relative to y. Higher values of k and y favor making transfers as do lower values of the expectation of t. This equilibrium does not permit the players to take advantage of their common interests. When t is high, it seems possible that the child would be able to signal this to his parent, permitting the parent to keep the food for herself. Indeed, it is possible to show that there exists another equilibrium outcome in which there is effective communication for interesting parameter values. If an equilibrium with two induced actions exists, there must be a cutoff type, t 1 (0, 1), such that t 1 is indifferent between receiving or not receiving the transfer, which defines t 1 = 1 k(1 y). Further, optimality of the parent s play requires that E[t t < t 1 ] y and E[t t > t 1 ] y. The latter inequality necessarily holds, since by simple algebra, t 1 y. Hence a two-step equilibrium exists if and only if E[t t < 1 k(1 y)] y. (11) This equilibrium is better for both parent and child than the babbling equilibrium. Hence in this situation effective cheap-talk communication is possible. In the equilibrium, the child can convey two different messages. When t is low, he says: I am very hungry, feed me. When t is high, he says: I am not hungry, keep the food for yourself. Clearly one can tell a similar story if the two strategic players are siblings rather than parent and child. Maynard Smith [6] and Lachmann and Bergstrom [5] show that if one adds handicaps to the basic model (so that a child with a low t finds it less expensive to call for food than a child with a high t), then there are also equilibria with costly signals. Deciding whether the model is a useful way to describe particular animal interactions is an empirical question. Presumably there are situations in which an animal s cries for food actually are costly (by making predators aware of the location of the animal, for example), but the example demonstrates that communication may be meaningful even when it is not costly. IV.2 Grunts and Gurneys Female Rhesus Monkeys use vocalizations to convey information about intention to attack or to behave in a conciliatory fashion. Cheney and Seyfarth [2] 10

12 describe these interactions. Silk, Kaldor, and Boyd [7] describes a strategic model of the behavior similar to the one that follows. These vocalizations apparently do not require much physical effort to produce so they are cheap in the sense that they require little energy to produce and may be in the repertoire of all monkeys. They also are quiet, so they probably do not impose additional costs like attracting potential predators. There are theoretical reasons to believe that effective cheap-talk communication is possible in this setting. Consider a situation in which there are two players. One player is a seated monkey, who will play the role of the receiver. The other player is a monkey who approaches the seated monkey. The approaching monkey may be tough or weak, representing her current strength. If permitted to do so, the weak money will approach the other one and groom her a cooperative gesture. This behavior is beneficial to both. One monkey gets groomed, while the other solidifies her position and may deter future aggression. On the other hand, the tough monkey intends to battle the seated monkey. The seated monkey must make a decision. If she stays, they she may be groomed or may face a battle. If she evades, they she will lose any grooming opportunity, but avoid a conflict. The table below provides suggestive payoffs for in four contingencies. Note that this is not a traditional payoff matrix: The columns represent strategies of the seated monkey (as is usual), but the rows represent the types of the approaching monkey, rather than her strategies. I have selected payoffs so that there is value in communication: The best outcome (for both players) is for the seated monkey to stay when the weak monkey approaches and to evade otherwise. I have justified the choice of payoffs for all but the tough monkey. The payoffs assume that the tough monkey would prefer to avoid a fight. With this formulation, the game satisfies the two necessary conditions for effective cheap talk. First, there is a conflict of interest between the two types of informed monkey: the tough monkey wants to encourage the other monkey to evade; the weak monkey wants to encourage the other monkey to stay. Second, there is common interest between the two players: the approaching monkey and the seated monkey want the seated monkey to stay in exactly the same situations. 11

13 Stay Evade Tough 1, 1 2, 0 Weak 1, 1 0, 0 Without communication, the seated monkey must decide what to do on the basis of a prior assessment about the right strategy. Staying is the right thing to do if the prior probability that the approaching monkey is weak is above.5. Now imagine that the approaching monkey can make sounds that will be interpreted correctly by the other monkey. The game now admits an equilibrium in which only the weak monkey makes these sounds, the seated monkey interprets these as signals that she will not face an attack. Otherwise, the seated monkey expects an attack and evades. For the given payoffs, it is optimal for only the weak monkey to make the sounds. The ability to communicate improves the outcome for everyone. In practice monkeys will have an idea about which other monkeys and weak and which are tough, and some conflict will be unavoidable to establish dominance relationships in a changing environment, the possibility of some form of effective cheap talk should persist. IV.3 Speculation Let me suggest other situations in which cheap talk might be effective. These comments are meant to be suggestive. In many species, colors change during the life cycle. At some level coloration will reflect fitness, but color may simply signal a bird s age rather than its fitness. Here the differentiation in color may make it easy to solve coordination problems. It is valuable to distinguish young from old. The young have nothing to gain from imitating sexually mature animals. The mature want to indicate that they are ready to mate. Rather than waste time and energy checking whether an individual is mature, it may be efficient to supply the information in the form of an easy to observe color change. Again, the two necessary conditions for effective cheap talk hold. First, there is common interest between the side that signals and the side that observes the signals. Both want to consider mating only if both are mature. Second, there is conflict of interest between the different kinds of signaler. Mature want to be seen as mature. Immature do not. 12

14 Another, even more behavior is flocking or migratory behavior. Here large groups of animals gain from coordination acting as a group. This creates the common interest that may make communication effective. If different members of the group are specialized (so that some are more able to lead the flock and others are unable to navigate correctly), then there may be reason to communicate to make sure that different individuals remain in best positions in the group. Dawkins and Guilford [4] describe color changes in male blue-headed wrasses that appear to satisfy the conditions for effective cheap-talk communication. When males are ready to spawn their body color changes from green to gray. By signaling their readiness to mate, males and females coordinate on the release of gametes. These examples describe interactions between members of the same species. Common interest across species can arise if there are sufficient gains for coordination. Coordination across time is a natural place to look for these differences of opinion. For example, different types of flower may be willing to bloom at different times in order to share pollinators. V Reconsidering Conflicts of Interest I argued that non-trivial cheap talk equilibria exist only when there is a conflict of interest between the Senders. This observation, as formalized by Proposition 1 is true, but perhaps misleading. It is misleading because all communication, whether cheap or costly, requires a conflict of interest between Senders. When signaling is expensive, the conflict comes because different types have different preferences over pairs of actions and messages (some peacocks are willing to produce a more colorful tail in order to gain reproductive success, while others are not). By looking at the problem more carefully you can see a divergence of preferences between types that is not immediately obvious. In the standard examples of the handicap, all Sender types want greater reproductive success. There is no conflict of interest between preferences over actions. There is a conflict when you imagine that preferences are two dimensional: dependent on both signal and action. So, for example, hungry children all prefer more food to less. In this respect, they all have the same interests. On the other hand, there may be a tradeoff between worrying about predators and worrying about whether you are going to starve. The hungrier the child, 13

15 the more willing he will be to risk attack by making a noisy signal to gain a bit more food. By expanding the set of things that matter to the Sender (adding dimensions to the set of actions), you can identify new reasons for communication. The preceding examples suggest that heterogeneous preferences arise when we introduce signaling costs. The added dimensional could be something else. If congestion is a problem, adding a time dimension may create the kind of conflict that permits effective signaling. Silk, Kaldor, and Boyd s [7] model of cheap communication between female Rhesus monkeys points out another source of common interest. They show that even if there is a conflict of interest in a one-shot interaction between monkeys, there may be sufficient common interest to make effective cheap talk credible in a repeated version of the game. For example, if the monkeys have common interest only when the signaler is weak, then it still may be that the expected value of coordination is positive, so that it is evolutionarily stable for a tough monkey to signal informatively in order to maintain the possibility of signaling informatively when weak. In order for this possibility to exist, types must not be persistent (so that being weak in one interaction does not mean that you will be weak in future interactions). VI On the Costs of Signals In this section I return to the standard case of costly signals and make a simple observation. In order for the handicap principal to work, it is not necessary for the signaler to make a large investment. Rather it is essential for would-be signalers to find it expensive to imitate the signal. A perfect signaling technology is one in which the fit type can costlessly signal fitness while the less fit type cannot signal at all. It is easy to think of examples in humans. A Hebrew speaker can cheaply signal her fluency in the language (at least to another Hebrew speaker) by introducing herself and describing the weather. Someone unfamiliar with the language could learn to imitate fixed statements (at a large cost), but could not pass a test of fluency. A musician can easily demonstrate his ability by picking up a violin and playing. The novice would be unable to imitate. If evolution operates on the signaling technology instead of just the choice of signals, then you might expect handicaps that impose no costs on those signaling, but are valuable nonetheless because they are costly to imitate. I do not know whether the 14

16 pressure to make signals less expensive for those who use them (and more expensive for those who do not) exists in the natural world. VII Conclusion I have argued that messages need not be expensive in order to be effective. By examining necessary conditions for effective cheap talk, we were led to realistic situations in which it is possible to transmit information without handicap. I concentrate on a particular kind of cheap talk in this essay: The Sender communicates about private information. Communication may also be a valuable way to describe intentions. In this setting, a message has the interpretation I am going to take an aggressive action rather than I am strong. Silk, Kaldor, and Boyd [7] gives examples of situations in which cheap communication about intentions can be effective in biological settings. I conclude by pointing out two limitations of my discussion. By concentrating on Nash equilibria, I examine necessary, but not sufficient condition for evolutionary stability. Confirming that informative equilibria are evolutionarily stable and describing the mechanism by which Receivers learn to interpret messages requires additional arguments. My analysis only suggest the possibility of effective cheap talk. The importance of the phenomenon is an empirical question. 15

17 References [1] Carl T. Bergstrom and Michael Lachmann. Signalling among relatives. III. Talk is cheap. Proceedings of the National Academy of Sciences, USA, 95: , [2] Dorothy L. Cheney and Robert M. Seyfarth. Reconciliatory grunts by dominant female baboons influence victims behaviour. Animal Behavior, 54: , [3] Vincent P. Crawford and Joel Sobel. Strategic information transmission. Econometrica, 50(6): , November [4] Marion S. Dawkins and Tim Guilford. Design of an intention signal in the bluehead wrasse (thalassoma bifasciatum). Proceedings of the Royal Society of London, Series B, 257: , [5] Michael Lachmann and Carl T. Bergstrom. Signalling among relatives. II. Beyond the tower of Babel. Theoretical Population Biology, 54: , [6] John Maynard Smith. Honest signalling: the Philip Sidney game. Animal Behavior, 42: , [7] Joan B. Silk, Elizabeth Kaldor, and Robert Boyd. Cheap talk when interests conflict. Animal Behavior, 59: , [8] Amos Zahavi. Mate selection- a selection for a handicap. Journal of Theoretical Biology, 53: ,